L(s) = 1 | − 3.46i·5-s + 3.46i·11-s − 5.19i·13-s + 6.92i·17-s + 7·19-s − 6.99·25-s + 5·31-s + 37-s + 10.3i·41-s − 1.73i·43-s + 6·47-s + 11.9·55-s − 18·65-s + 1.73i·67-s − 3.46i·71-s + ⋯ |
L(s) = 1 | − 1.54i·5-s + 1.04i·11-s − 1.44i·13-s + 1.68i·17-s + 1.60·19-s − 1.39·25-s + 0.898·31-s + 0.164·37-s + 1.62i·41-s − 0.264i·43-s + 0.875·47-s + 1.61·55-s − 2.23·65-s + 0.211i·67-s − 0.411i·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.087530992\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.087530992\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 5.19iT - 13T^{2} \) |
| 17 | \( 1 - 6.92iT - 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 + 1.73iT - 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 1.73iT - 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 - 8.66iT - 73T^{2} \) |
| 79 | \( 1 - 15.5iT - 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.040187477939513771311360637997, −7.44398135024813468885859670508, −6.39155363510499330159414877566, −5.58821386465766116908128869490, −5.13075357967502039251074227677, −4.41389453316476277063711347858, −3.66760424636713438528294124816, −2.64004020004063021094859250249, −1.47399982517872016891569393041, −0.871342897843147157232852418468,
0.67987550782813973866333310147, 2.05546043617351061551266639194, 2.94111092834589446620875486245, 3.33429489693943252873393223599, 4.31562701063703723150733037427, 5.23699617192387628994069607111, 6.01463174764086440807617711324, 6.67815663342761930656247052334, 7.26697735051634026350701299758, 7.65611881761990967708807069693